On the Khovanov and knot Floer homologies of quasi-alternating links

TL;DR

This paper proves that quasi-alternating links are homologically thin in Khovanov and knot Floer homologies, with their homology groups determined by classical invariants like signature and polynomials.

Contribution

It establishes the homological thinness of quasi-alternating links for both Khovanov and knot Floer homologies, linking these homologies to classical link invariants.

Findings

Quasi-alternating links are homologically thin in Khovanov and knot Floer homologies.

Homology groups are determined by signature and polynomial invariants.

Uses exact triangles relating link homologies at crossings.

Abstract

Quasi-alternating links are a natural generalization of alternating links. In this paper, we show that quasi-alternating links are "homologically thin" for both Khovanov homology and knot Floer homology. In particular, their bigraded homology groups are determined by the signature of the link, together with the Euler characteristic of the respective homology (i.e. the Jones or the Alexander polynomial). The proofs use the exact triangles relating the homology of a link with the homologies of its two resolutions at a crossing.